Energy Stable Simulation of Two-Phase Equilibria with ... · Energy Stable Simulation of Two-Phase...

Energy Stable Simulation of Two-PhaseEquilibria with Capillarity?

Shuyu Sun1[0000−0002−3078−864X]

Computational Transport Phenomena Laboratory, Division of Physical Science andEngineering, King Abdullah University of Science and Technology, Thuwal

23955-6900, Kingdom of Saudi Arabia. [email protected]://web.kaust.edu.sa/faculty/shuyusun/

Abstract. We consider the affect of capillary pressure on the Van derWaals fuid and on the Peng-Robinson fluid by minimizing total Helmholtzenergy in given total volume, temperature, and total moles. We proposesimple but conditionally energy stable numerical schemes, and we pro-vide interesting numerical examples. We compare our numerical resultswith the prediction of Kelvin’s equation, indicating that Kelvin’s equa-tion works well only when the temperature is not too low.

Keywords: Interfacial tension · The Kelvin equation · Phase diagrams· Peng-Robinson equation of state · Van der Waals equation of state.

1 Introduction

Two-phase and multi-phase flows are important and common phenomena inpetroleum industry, where oil, gas and water are often produced and transportedtogether. In particular, engineers and researchers in reservoir engineering studydrainage problems arising during the development and production of oil and gasreservoirs so as to obtain a high economic recovery, by developing, conducting,and interpolating the simulation of subsurface flows of reservoir fluids, includingwater, hydrocarbon, CO2, H2S for example in porous geological formation. Field-scale (Darcy-scale) simulation has conventionally and routinely used for thispurpose [12, 4, 17, 16, 2]. A number of parameters like relative permeability andcapillary pressure are taken as given functions in Darcy-scale simulation [14]. Tostudy these parameters as well as to obtain deep understanding of porous mediaflow and transport, researchers develop and utilize pore-scale simulation of two-phase [1, 11, 6, 9, 8, 7, 5, 13, 3], which has been shown to be a great research toolto understand the complex hydrodynamic and phase behaviors of the systems.

? This work was supported by funding from King Abdullah University of Science andTechnology (KAUST) through grants BAS/1/1351-01.

ICCS Camera Ready Version 2019To cite this paper please use the final published version:

DOI: 10.1007/978-3-030-22747-0_40

https://dx.doi.org/10.1007/978-3-030-22747-0_40

2 S. Sun

2 Mathematical Modeling Framework

2.1 Mathematical Model of Bulk Properties

In this paper we consider the Van der Waals equation of state (EOS) and thePeng-Robinson EOS [15] to model the bulk properties of our fluid system. Wenote that the Van der Waals EOS is one of the simplest equations of state thatallow phase splitting. On the other hand, the Peng-Robinson EOS is the mostpopular equation of state (EOS) to model and compute the fluid equilibriumproperty of hydrocarbon fluid and other petroleum fluids, and it is widely usedin reservoir engineering and oil industries. Some of the material in this sectioncan be found in many textbooks, but we list them here for completeness ofinformation only.

The PVT-form of the Van der Waals EOS appears as(p+

a

v2

)(v − b) = RT. (1)

At the critical condition, we have(∂p∂v

)T= 0 and

(∂2p∂v2

)T= 0. These two con-

ditions together with the above Van der Waals EOS lead to vc = 3b, pc = a27b2 ,

Tc = 8a27Rb , and Zc = 3

8 . we substitute the definitions of the reduced pressure,reduced molar volume and reduced temperature with the values of critical prop-erties (in terms of a and b) into the above Van der Waals EOS to obtain thefollowing reduced form of the Van der Waals equation(

pr +3

v2r

)(3vr − 1) = 8Tr. (2)

From the PVT-form of the Van der Waals EOS, we can derive the bulkHelmholtz free energy for the Van der Waals fluid as F (T, V,N) = Fideal(T, V,N)+Fexcess(T, V,N), where the ideal gas contribution (note that N =

∑iNi) is

Fideal(T, V,N) = RT∑Mi=1Ni ln

(NiV

)+ NCintg(T ), with one choice of Cintg as

Cintg(T ) = −RT(ln(

nQNA

) + 1)

and nQ =(2πmkBT

h2

)3/2. The excess part of

Helmholtz free energy is: Fexcess(T, V,N) = −NRT ln(1− Nb

V

)− N2a

V .We define the Helmholtz free energy density f as the Helmholtz free energy

per unit volume of fluid. It is clear that

f(T,n) = fideal(T,n) + fexcess(T,n), (3)

fideal(T,n) = RT

M∑i=1

ni ln (ni) + nCintg(T ), (4)

fexcess(T, n) = −nRT ln (1− bn)− an2. (5)

With the Peng-Robinson EOS, the bulk Helmholtz free energy density f(T,n)of a bulk fluid is determined by f(T,n) = fideal(T,n) + fexcess(T,n) and

f ideal(T,n) = RT

M∑i=1

ni (lnni − 1) , (6)


DOI: 10.1007/978-3-030-22747-0_40

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Two-Phase Equilibria with Capillarity 3

f excess(T, n) = −nRT ln (1− bn) + a(T )n

2√2b

ln

(1 + (1−

√2)bn

1 + (1 +√2)bn

). (7)

The two parameters a and b can be computed as follows. For a mixture, theseparameters can be calculated from the ones of the pure fluids by mixing rules:

a(T ) =

M∑i=1

M∑j=1

yiyjaia1/2j (1− kij), b =

M∑i=1

yibi, (8)

where yi = ni/n is the mole fraction of component i, and ai and bi are thePeng-Robinson parameters for pure-substance component i. We often use exper-imental data to fit the binary interaction coefficient kij of Peng-Robinson. Forconvenience, kij is usually assumed to be constant for a fixed species pair.

Even though the pure-substance Peng-Robinson parameters ai and bi canalso be fit by using experimental data, they can also be computed from thecritical properties of the species:

ai = ai (T ) = 0.45724R2T 2

ci

Pci

(1 +mi

(1−

√T

Tci

))2

,

bi = 0.07780RTciPci

.

As intrinsic properties of the species, the critical temperature Tci and criticalpressure Pci of a pure substance are available for most substances encounteredin engineering practice. In the above formula for ai, we need also to specifythe parameter mi for modeling the influence of temperature on ai. It was sug-gested that one may correlates the parameter mi experimentally to the accentricparameter ωi of the species by the following equations:

mi = 0.37464 + 1.54226ωi − 0.26992ω2i , ωi ≤ 0.49,

mi = 0.379642 + 1.485030ωi − 0.164423ω2i + 0.016666ω3

i , ωi > 0.49.

The accentric parameter can be fit by using experimental data, but if we lackdata, we can also compute it by using critical temperature Tci , critical pressurePci and the normal boiling point Tbi :

ω =3

7

log10

(Pci

14.695 PSI

)TciTbi− 1

− 1 =3

7

log10

(Pci

1 atm

)TciTbi− 1

− 1.

Based on the fundamental relation on thermodynamic variables, the pressureof homogeneous fluids p and the Helmholtz free energy f(n) can be linked in thefollowing way

p = p(n, T ) = −(∂F (n, T,Ω)

∂V

)T,N

= −

(∂(f(NV , T

)V)

∂V

)T,N


DOI: 10.1007/978-3-030-22747-0_40

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4 S. Sun

= −f − VM∑i=1

(∂f

∂ni

)T,n1,···,ni−1,ni+1,···nM

(∂NiV∂V

)Ni

=

M∑i=1

ni

(∂f

∂ni

)T,n1,···,ni−1,ni+1,···nM

− f =

M∑i=1

niµi − f.

Substitution of the Peng-Robinson expression of f leads to

p =nRT

1− bn− n2a(T )

1 + 2bn− b2n2=

RT

v − b− a(T )

v(v + b) + b(v − b).

2.2 Modeling the Two-Phase Systems with Interfaces

The total Helmholtz energy F tot has two contributions, one from the homoge-neous bulk fluid, and another one from the interface between the two phases:

F tot (n) = Fbulk (n; T,Ω) + Finterface (n; T,AI)

=

ˆΩ\AI

f(n; T )dx+

ˆAI

σ([n], n; T )ds,

where σ is the interfacial tension (or the interfacial Helmholtz energy per unitarea), which is a function of the jump and average of n across the interface.In the paper, for convenience, we assume that σ is a given constant. With thisassumption, we have

F tot = f(nL)V L + f

(nG)V G + σAI ,

where nL and V L are the molar density and volume of liquid phase, respectively.Meanings of nG and V G are similar to these.

We impose the total volume V tot and the total moles Ntot such as V tot =V L + V G and N tot

i = nLi VL + nGi V

G, i = 1, 2, · · · ,M. The interface AI is also afunction of V G; for example, if there is only one single bubble, then AI = 4πr2

while V G = 4πr3

3 . Considering nG and V G as the primary variables, we can writethe total Helmholtz energy F tot as

F tot(nG, V G) = f

(Ntot − nGV G

V tot − V G

)(V tot − V G) + f

(nG)V G + σAI(V

G).

When F tot achieves its minimum and the function F tot is smooth, we have

∂F tot(nG, V G)

∂nGi= 0, and

∂F tot(nG, V G)

∂V G= 0,

which implies respectively µ(nG) = µ(nL), and pG − pL = σ dAIdV G

. If there isonly one single bubble, then AI = 4πr2 and V G = 4πr3

3 ; the above conditionsimplifies to

pG − pL =2σ

r,

which is known as the Young-Laplace equation.


DOI: 10.1007/978-3-030-22747-0_40

https://dx.doi.org/10.1007/978-3-030-22747-0_40


3 Numerical Methods

To find the minimum of the F tot(nG, V G), we design the following ordinarydifferential equation (ODE) system

∂NGi

∂t= −∂N

Li

∂t= kNiN

toti (µ(nL)− µ(nG)), i = 1, 2, · · · ,M,

∂V G

∂t= −∂V

L

∂t= kV V

tot(pG(nG)− pL(nL)− σ dAIdV G

).

We solve the above ODE using the following explicit Euler method, which isconditionally stable provided that the time step is less than a certain value.

NG,k+1i −NG,k

i

tk+1 − tk= kNiN

toti (µ(nL,k)− µ(nG,k)), i = 1, 2, · · · ,M,

V G,k+1 − V G,k

tk+1 − tk= kV V

tot(pG(nG,k)− pL(nL,k)− σ dAIdV G

),

where nG,k and nL,k can be determined from V G,k, NG,ki , i = 1, 2, · · · ,M.

Existence, uniqueness, and energy-decay property of the solution to the aboveODE equation system as well as the existence, uniqueness, and the conditionalenergy-decay property of the numerical solution defined above can be provedusing techniques similar to the ones used in our previous work [10].

4 Numerical Examples

4.1 Effect of Capillary Pressure on the Van der Waals Fluid

We first consider the single-component two-phase fluid system modeled by theVan der Waals EOS. If we write the single-component Van der Waals EOSusing reduced temperature, reduced pressure and reduced molar volume, wethen may obtain an universal dimensionless equation (2) for the EOS. Thatis, all single-component two-phase fluids behave in the same way after certainlinear transformation. Since we will report results in reduced quantities, it doesnot matter the parameters a and b we choose, but in the implementation, thechoice of kN and kV might depend on the specific values of a and b. Withoutloss of generality, we choose a = 3 and b = 1

3 ; in this way, pc = 1 and vc = 1 andthus pr = p and vr = v. The units of these quantities can be any fixed units, aslong as consistent units are used, such as the SI unit system.

For comparison and verification, we first let σ = 0 in our model, and wecalculate the liquid-vapor phase behavior of the Van der Waals fluid withoutcapillarity. We generate phase diagrams numerically and plot them in Figures1(a) and 1(b). Figure 1(a) is the volume-temperature phase envelope while Figure1(b) displays reduced boiling temperature as a function of reduced pressure.

We then consider two cases with capillary pressure. The first case is a sin-gle bubble of radius r immersed in the liquid. In this case, V G = 4πr3

3 and


DOI: 10.1007/978-3-030-22747-0_40

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6 S. Sun

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

reduced molar density

redu

ced

tem

pera

ture

1e−08 1e−06 1e−04 1e−02 1e+00

0.0

0.2

0.4

0.6

0.8

1.0

reduced pressure

redu

ced

tem

pera

ture

(a) (b)

Fig. 1. Phase diagrams of the single-component Van der Waals fluid: (a) reduced molarvolumes of gas and liquid phases at equilibrium as a function of reduced temperature;(b) reduced boiling temperature as a function of reduced pressure

dAI/dVG = 2/r. For this single-component two-phase fluid system, our model-

ing ODE reduces to

∂NG

∂t= kNN

tot(µ(nL)− µ(nG)),

∂r

∂t=kV V

4πr2

tot

(pG(nG)− pL(nL)− 2σ

r).

The values of kN and kV are manually tuned for one typical simulation, andare then fixed for all other runs. In all numerical examples in this subsection,we choose kN = 0.05 and kV = 0.02. We use the unit time step ∆t = 1 for allnumerical runs in this paper.

The effect of capillary pressure on the saturation pressure of the liquid phaseand the vapor (gas) phase is provided in Figures 2(a)-2(d) under various con-dition of reduced temperatures and capillary pressures. The horizontal axis ofthese plots are σ

rpc. The ratio of interfacial tension to the radius of the gas bubble

σr is the influencing factor, we divide this ratio by the critical pressure of thefluid pc so that we get a dimensionless quantity.

We also compare the our numerical prediction of vapor pressure with theone calculated by the Kelvin equation. The Kelvin equation can be derived byusing the two equilibrium conditions dpG − dpL = d(2σ/r) and dµG = dµL,and the two fundamental relations dµG = vGdpG and dµL = vLdpL, which yielddpG =

d( 2σr )

(1− vGvL

). By assuming the ideal gas law for the vG and assuming a constant

molar density for vL, we can integrate the above differential equation to obtain


DOI: 10.1007/978-3-030-22747-0_40

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0.000 0.005 0.010 0.015

0.16

0.17

0.18

0.19

0.20

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.18

00.

185

0.19

00.

195

0.20

0

redu

ced

pres

sure

of g

as p

hase

Pr_liquid (left axis)

Pr_gas (right axis)

Pr_gas from Kelvin's equation (right axis)

0.000 0.005 0.010 0.0150.

340.

350.

360.

370.

38

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.36

50.

370

0.37

50.

380

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


(a) (b)

0.000 0.005 0.010 0.015

0.60

0.61

0.62

0.63

0.64

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.62

50.

630

0.63

50.

640

0.64

5

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


0.000 0.005 0.010 0.015 0.020

0.75

0.76

0.77

0.78

0.79

0.80

0.81

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.78

00.

785

0.79

00.

795

0.80

00.

805

0.81

0

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


(c) (d)

Fig. 2. The effect of capillary pressure on the saturation pressures of the liquid phaseand the vapor (gas) bubble modeled by the single-component Van der Waals EOS: (a)Tr = 0.7 (b) Tr = 0.8 (c) Tr = 0.9 (d) Tr = 0.95


DOI: 10.1007/978-3-030-22747-0_40

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8 S. Sun

0.000 0.005 0.010 0.015

0.12

60.

127

0.12

80.

129

0.13

00.

131

0.13

2

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

2.13

42.

136

2.13

82.

140

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

0.000 0.005 0.010 0.0150.

236

0.23

80.

240

0.24

20.

244

0.24

60.

248

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

1.92

01.

925

1.93

0

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

(a) (b)

0.000 0.005 0.010 0.015

0.41

0.42

0.43

0.44

0.45

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

1.61

1.62

1.63

1.64

1.65

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

0.000 0.005 0.010 0.015 0.020

0.55

0.60

0.65

0.70

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

1.20

1.25

1.30

1.35

1.40

1.45

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

(c) (d)

Fig. 3. The effect of capillary pressure on the molar densities of the liquid phase andthe vapor (gas) bubble modeled by the single-component Van der Waals EOS: (a)Tr = 0.7 (b) Tr = 0.8 (c) Tr = 0.9 (d) Tr = 0.95


DOI: 10.1007/978-3-030-22747-0_40

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0.000 0.005 0.010 0.015

0.20

0.21

0.22

0.23

0.24

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.20

00.

205

0.21

00.

215

0.22

0

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


0.000 0.005 0.010 0.0150.

390.

400.

410.

42

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.38

50.

390

0.39

50.

400

0.40

5

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


(a) (b)

0.000 0.005 0.010 0.015

0.65

0.66

0.67

0.68

0.69

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.65

00.

655

0.66

00.

665

0.67

0

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


0.000 0.005 0.010 0.015

0.81

0.82

0.83

0.84

0.85

0.86

0.87

sigma/r/Pc

redu

ced

pres

sure

of l

iqui

d ph

ase

0.81

50.

820

0.82

50.

830

0.83

50.

840

redu

ced

pres

sure

of g

as p

hase


Pr_gas (right axis)


(c) (d)

Fig. 4. The effect of capillary pressure on the saturation pressures of the liquid dropletand the vapor (gas) phase modeled by the single-component Van der Waals EOS: (a)Tr = 0.7 (b) Tr = 0.8 (c) Tr = 0.9 (d) Tr = 0.95


DOI: 10.1007/978-3-030-22747-0_40

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10 S. Sun

0.000 0.005 0.010 0.015

0.12

80.

129

0.13

00.

131

0.13

20.

133

0.13

4

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

2.13

82.

140

2.14

22.

144

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

0.000 0.005 0.010 0.0150.

240

0.24

20.

244

0.24

60.

248

0.25

00.

252

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

1.92

51.

930

1.93

51.

940

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

(a) (b)

0.000 0.005 0.010 0.015

0.43

0.44

0.45

0.46

0.47

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

1.64

1.65

1.66

1.67

1.68

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

0.000 0.005 0.010 0.015

0.58

0.60

0.62

0.64

0.66

0.68

0.70

0.72

sigma/r/Pc

redu

ced

mol

ar d

ensi

ty o

f gas

pha

se

1.40

1.42

1.44

1.46

1.48

1.50

1.52

redu

ced

mol

ar d

ensi

ty o

f liq

uid

phas

e

vc*n (gas)

vc*n (liquid)

(c) (d)

Fig. 5. The effect of capillary pressure on the molar densities of the liquid dropletand the vapor (gas) phase modeled by the single-component Van der Waals EOS: (a)Tr = 0.7 (b) Tr = 0.8 (c) Tr = 0.9 (d) Tr = 0.95


DOI: 10.1007/978-3-030-22747-0_40

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0 2000 4000 6000 8000 10000

9250

0093

0000

9350

0094

0000

9450

00

sigma/r (Pa)

pres

sure

of l

iqui

d ph

ase

(Pa)

9455

0094

6000

9465

0094

7000

9475

00

pres

sure

of g

as p

hase

(P

a)

liquid pressure (left axis)

gas pressure (right axis)

gas pressure from Kelvin's equation (right axis)

0 2000 4000 6000 8000 10000

403.

040

3.5

404.

040

4.5

405.

040

5.5

sigma/r (Pa)

mol

ar d

ensi

ty o

f gas

pha

se (

mol

/m^3

)

8874

8875

8876

8877

8878

8879

mol

ar d

ensi

ty o

f liq

uid

phas

e (m

ol/m

^3)

gas molar density

liquid molar density

(a) (b)

Fig. 6. The effect of capillary pressure on phase behaviors of a vapor (gas) bubble im-mersed in its saturated liquid phase modeled by the single-component Peng-RobinsonEOS: (a) liquid and vapor (gas) pressures (b) molar densities of two phases

0 2000 4000 6000 8000 10000

9500

0095

5000

9600

0096

5000

9700

00

sigma/r (Pa)

pres

sure

of l

iqui

d ph

ase

(Pa)

9480

0094

8500

9490

0094

9500

pres

sure

of g

as p

hase

(P

a)

liquid pressure (left axis)

gas pressure (right axis)

gas pressure from Kelvin's equation (right axis)

0 2000 4000 6000 8000 10000

404

405

406

407

sigma/r (Pa)

mol

ar d

ensi

ty o

f gas

pha

se (

mol

/m^3

)

8875

8876

8877

8878

8879

8880

8881

mol

ar d

ensi

ty o

f liq

uid

phas

e (m

ol/m

^3)

gas molar density

liquid molar density

(a) (b)

Fig. 7. The effect of capillary pressure on phase behaviors of a liquid droplet immersedin its vapor (gas) phase modeled by the single-component Peng-Robinson EOS: (a)liquid and vapor (gas) pressures (b) molar densities of two phases


DOI: 10.1007/978-3-030-22747-0_40

https://dx.doi.org/10.1007/978-3-030-22747-0_40

12 S. Sun

the following Kelvin equation

pG = p∞ exp(−2σvL

rRT),

which quickly reveals that the vapor pressure decreases with increasing interfacecurvature and increasing interfacial tension.

Figures 2(a)-2(d) indicates that the vapor pressure predicted by the Kelvinequation agree well with our numerical simulation when the temperature is nottoo low. When the temperature is low, the vapor pressure predicted by theKelvin equation has pronounced derivation, because the ideal gas law is no lonera good approximation at low temperature. From Figures 2(a)-2(d), we also seethat the saturation liquid pressure departs from its zero-capillary value muchmore significantly than the saturation vapor pressure.

We observe the number of time steps required for convergence varies withthe reduced temperature. When Tr = 0.7, the number of time steps requiredfor convergence is about 10 to 30 depending on the value of σ

rpc, while when

Tr = 0.95, the number of time steps required for convergence is can be morethan 2000.

In Figures 3(a)-3(d), we show the variation of molar densities of the satu-rated liquid phase and saturated vapor phase as influenced by capillarity. It isevident that the molar densities of both phases decrease with increasing interfacecurvature and increasing interfacial tension. This is likely because the decreaseof saturation pressure causes the decrease of molar densities.

The second case is a single liquid droplet of radius r immersed in the vaporphase. In this case, V G = V tot − 4πr3

3 and dAI/dVG = −2/r. For this single-

component two-phase fluid system, our modeling ODE reduces to

∂NG

∂t= kNN

tot(µ(nL)− µ(nG)),

∂r

∂t=kV V

4πr2

tot

(pL(nL)− pG(nG)− 2σ

r).

Figures 4(a)-4(d) provide the trend of the saturation liquid pressure of a sin-gle liquid droplet and the surrounding saturation vapor pressure as influencedby capillarity. Unlike the vapor bubble case, both the liquid pressure and thesaturation vapor pressure increase here with increasing interface curvature andincreasing interfacial tension. The trend of the saturation vapor pressure is alsocompared with the ones predicted by the Kelvin equation, which does a goodjob only when the temperature is not too low. In Figures 5(a)-5(d), we show thevariation of molar densities of the saturated liquid phase and saturated vaporphase as influenced by capillarity. Unlike the vapor bubble case, the molar den-sities of both phases increase with increasing interface curvature and increasinginterfacial tension.


DOI: 10.1007/978-3-030-22747-0_40

https://dx.doi.org/10.1007/978-3-030-22747-0_40


Table 1. Relevant data of isobutane (excerpted from Table 3.1 of [4], Page 141)

component symbol Tc, K Pc, MPa Tb, Kisobutane nC4 425.18 3.797 272.64

4.2 Effect of Capillary Pressure on the Peng-Robinson Fluid

The Peng-Robinson EOS, unlike the single-component Van der Waals EOS, can-not transformed into a same reduced from by using reduced temperature, reducedpressure and reduced molar volume, because a third parameter, the accentric pa-rameter appears in the Peng-Robinson EOS. We thus consider a specific examplebelow with physical units. In the example below, we consider the species of isobu-tane (nC4) at the temperature of 350 K. For convenience of readers, we providein Table 1 its critical properties and its normal boiling point. The values of kNand kV are manually tuned for one typical simulation, and are then fixed for allother runs. In all runs in this subsection, we choose kN = 1 × 10−3 mol J−1 s−1

and kV = 1× 10−9 Pa−1 s−1. We again use the unit time step ∆t = 1 s.Figure 6(a) displays the effect of capillary pressure on the saturation pressures

of both phases when a single gas bubble is in equilibrium on its liquid phase. Thecorresponding variation of molar densities of both phases are plotted in Figure6(b). The trend of the saturation vapor pressure is also compared with the onespredicted by the Kelvin equation, which does a reasonable job in the conditionsimulated in this example. Clearly, like the case modeled by the Van der WaalsEOS, both the liquid pressure and the saturation vapor pressure decrease herewith increasing interface curvature and increasing interfacial tension, which alsoleads a decrease of molar density of both phases.

Figure 7(a) and Figure 7(b) show the variation of the saturation pressuresand molar densities of both phases when a single liquid droplet is in equilibriumon its vapor phase under various conditions of interface curvature and increasinginterfacial tension. The trend seems similar to the case predicted by the single-component Van der Waals EOS qualitatively.

5 Conclusion

In this paper, we formulate a framework to model a two-phase fluid with a(sharp) interface using Van der Waals EOS and Peng-Robinson EOS. Our modelis able to predict the affect of capillary pressure on phase behaviors. We proposesimple but conditionally energy stable numerical schemes to solve the minimiza-tion problem of total Helmholtz energy in given total volume, temperature, andtotal moles. We consider bubbles in liquid as well as droplets in vapor in ournumerical examples. Our numerical agree well with the prediction of Kelvin’sequation when the temperature is not too low. Due to the limitation of thespace, we provide only single-component examples. Our model and schemes,however, are expected to work well with multi-component systems, which willbe investigated and reported in a separate paper elsewhere.


DOI: 10.1007/978-3-030-22747-0_40

https://dx.doi.org/10.1007/978-3-030-22747-0_40

14 S. Sun

References

1. Breure, B., Peters, C.: Modeling of the surface tension of pure components andmixtures using the density gradient theory combined with a theoretically derivedinfluence parameter correlation. Fluid Phase Equilibria (2012)

2. Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow andtransport. Computer Methods in Applied Mechanics and Engineering 193, 2565–2580 (2004)

3. Fan, X., Kou, J., Qiao, Z., Sun, S.: A component-wise convex splitting scheme fordiffuse interface models with van der waals and peng-robinson equations of state.SIAM Journal on Scientific Computing 39(1), B1–B28 (2017)

4. Firoozabadi, A.: Thermodynamics of hydrocarbon reservoirs. McGraw-Hill, NewYork (1999)

5. Kou, J., Sun, S.: Multi-scale diffuse interface modeling of multi-component two-phase flow with partial miscibility. Journal of Computational Physics 318, 349–372(2016)

6. Kou, J., Sun, S.: Entropy stable modeling of non-isothermal multi-componentdiffuse-interface two-phase flows with realistic equations of state. Computer Meth-ods in Applied Mechanics and Engineering 341, 221–248 (2018)

7. Kou, J., Sun, S.: A stable algorithm for calculating phase equilibria with capillarityat specified moles, volume and temperature using a dynamic model. Fluid PhaseEquilibria 456, 7–24 (2018)

8. Kou, J., Sun, S.: Thermodynamically consistent modeling and simulation of multi-component two-phase flow model with partial miscibility. Computer Methods inApplied Mechanics and Engineering 331, 623–649 (2018)

9. Kou, J., Sun, S.: Thermodynamically consistent simulation of nonisothermaldiffuse-interface two-phase flow with peng-robinson equation of state. Journal ofComputational Physics 371, 581–605 (2018)

10. Kou, J., Sun, S., Wang, X.: An energy stable evolutional method for simulatingtwo-phase equilibria of multi-component fluids at constant moles, volume and tem-perature. Computational Geosciences 20(1), 283–295 (2016)

11. Kou, J., Sun, S., Wang, X.: Linearly decoupled energy-stable numerical methodsfor multi-component two-phase compressible flow. SIAM Journal on NumericalAnalysis 56(6), 3219–3248 (2018)

12. Lake, L.W.: Enhanced oil recovery. Prentice Hall, Englewood Cliffs, New Jersey(1989)

13. Li, Y., Kou, J., Sun, S.: Numerical modeling of isothermal compositional gradingby convex splitting methods. Journal of Natural Gas Science and Engineering 43,207–221 (2017)

14. Moortgat, J., Sun, S., Firoozabadi, A.: Compositional modeling of three-phase flowwith gravity using higher-order finite element methods. Water Resources Research47, W05511 (2011)

15. Peng, D.Y., Robinson, D.: A new two-constant equation of state. Industrial andEngineering Chemistry Fundamentals 15(1), 59–64 (1976)

16. Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous galerkin meth-ods for reactive transport in porous media. SIAM Journal on Numerical Analysis43(1), 195–219 (2005)

17. Sun, S., Wheeler, M.F.: Local problem-based a posteriori error estimators fordiscontinuous galerkin approximations of reactive transport. Computational Geo-sciences 11(2), 87–101 (2007)


DOI: 10.1007/978-3-030-22747-0_40

https://dx.doi.org/10.1007/978-3-030-22747-0_40

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